High Energy Seminar: "Large N Tensor Models"

Mon, 04/10/2017 - 14:00

Igor Klebanov (Princeton University)

We review the double line
notation for the Feynman diagram expansion of N by N matrix models. In the ‘t
Hooft large N limit only the planar diagrams survive, and the dual graphs may
be thought of as discretized random surfaces. We proceed to theories where the
dynamical degrees of freedom are rank-3 tensors with distinguishable indices,
each of which takes N values. Their Feynman diagrams may be drawn using colored
triple lines (red, blue, green), while the dual graphs are made out of
tetrahedra glued along their triangular faces. Such theories possess a special
solvable large N limit dominated by the “melon” diagrams. We discuss quantum
mechanical models of fermionic rank-3 tensors and their similarity with the
Sachdev-Ye- Kitaev disordered model. We then use the large N Schwinger-Dyson
equations to study the conformal dimensions of certain composite operators.
Gauging the global symmetry in the quantum mechanical models removes the
non-singlet states; therefore, one can search for their well-defined gravity
duals. We note that the models possess a vast number of gauge-invariant
operators involving higher powers of the tensor field. Finally, we discuss
similar models of a commuting rank-3 tensor in dimension d. While the quartic
interaction is not positive definite, we study the large N Schwinger-Dyson
equations and show that their solution is consistent with conformal invariance.